From the perspective of an investor, most of the portfolios in the investment opportunity set (the set of possible portfolios) are irrelevant. An investor who wants to maximize portfolio performance is interested only in undominated portfolios.

You can find the set of undominated portfolios by examining the investment opportunity set in Figure 4.2. You know that an undominated portfolio must satisfy one of two criteria:

1. For a target expected return, there must exist no other portfolio that provides less risk, or

2. For a target level of risk there must exist no other portfolio that provides a higher expected return.

You can verify that these criteria lead to the heavy line extending northeast from the vertex in Figure 4.2. This line is called the efficient frontier.

Finding the efficient portfolios by varying portfolio weights is like "looking for a needle in a haystack." Here is where mathematics comes to the rescue. If we can construct an analytical solution for the efficient frontier, we can solve for any point on this frontier. This process is started in the next topic titled Two-Firm Analysis.

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